Abstract
We consider the problem of finding a maximal matching of minimum size, given an unweighted general graph. This problem is a well studied and it is known to be NP-hard even for some restricted classes of graphs. Moreover, in case of general graphs, it is NP-hard to approximate the Minimum Maximal Matching (shortly MMM) within any constant factor smaller than \(\frac{7}{6}\). The current best known approximation algorithm is the straightforward algorithm which yields an approximation ratio of 2. We propose the first nontrivial algorithm yields an approximation ratio of \(2 - c \frac{\log{n}}{n}\), for an arbitrarily positive constant c. Our algorithm is based on the local search technique and utilizes an approximate solution of the Minimum Weighted Maximal Matching problem in order to achieve the desirable approximation ratio.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41, 153–180 (1994)
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–46 (1985)
ChlebÃk, M., ChlebÃkova, J.: Approximation hardness of edge dominating set problems. Journal of Combinatorial Optimization 11(3), 279–290 (2006)
Carr, R., Fujito, T., Konjevod, G., Parekh, O.: A \(2\frac{1}{10}\) -approximation algorithm for a generalization of the weighted edge-dominating set problem. Journal of Combinatorial Optimization 5(3), 317–326 (2001)
Cardinal, J., Langerman, S., Levy, E.: Improved Approximation Bounds for Edge Dominating Set in Dense Graphs. In: Erlebach, T., Kaklamanis, C. (eds.) WAOA 2006. LNCS, vol. 4368, pp. 108–120. Springer, Heidelberg (2007)
Fujito, T., Nagamochi, H.: A 2-Approximation Algorithm for the Minimum weight edge dominating set problem. Discrete Applied Mathematics 181, 199–207 (2002)
Garey, M.R., Johnson, D.S.: Computers and intractability. A guide to the theory of NP-completeness. W. H. Freeman and Co., New York (1979)
Halperin, E.: Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. SIAM Journal on Computing 31(5), 1608–1623 (2002)
Harary, F.: Graph Theorey. Addison-Wesley, Reading (1969)
Horton, J.D., Kilakos, K.: Minimum edge dominating sets. SIAM Journal on Discrete Mathematics 6(3), 375–387 (1993)
Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs. Journal of Algorithms 26(2), 238–274 (1998)
Srinivasan, A., Madhukar, K., Nagavamsi, P., Rangan, C.P., Chang, M.S.: Edge domination on bipartite permutation graphs and cotriangulated graphs. Information Processing Letters 56(3), 165–171 (1995)
Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM Journal on Applied Mathematics 38, 364–372 (1980)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gotthilf, Z., Lewenstein, M., Rainshmidt, E. (2009). A \((2 - c \frac{\log {n}}{n})\) Approximation Algorithm for the Minimum Maximal Matching Problem. In: Bampis, E., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2008. Lecture Notes in Computer Science, vol 5426. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93980-1_21
Download citation
DOI: https://doi.org/10.1007/978-3-540-93980-1_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-93979-5
Online ISBN: 978-3-540-93980-1
eBook Packages: Computer ScienceComputer Science (R0)